Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle MON = 2x + 29$, and $ m \angle LOM = 3x - 34$, find $m\angle MON$. $O$ $L$ $N$ $M$
Explanation: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {3x - 34} + {2x + 29} = {90}$ Combine like terms: $ 5x - 5 = 90$ Add $5$ to both sides: $ 5x = 95$ Divide both sides by $5$ to find $x$ $ x = 19$ Substitute $19$ for $x$ in the expression that was given for $m\angle MON$ $ m\angle MON = 2({19}) + 29$ Simplify: $ {m\angle MON = 38 + 29}$ So ${m\angle MON = 67}$.